Preheat oven to 425°. Melt butter in a large nonstick skillet over medium-high heat; add onion and next 3 ingredients. Cook, stirring occasionally, 8 minutes or until tender.
Unroll piecrust onto a lightly floured surface. Pat or roll into a 12-inch circle. Place piecrust on a parchment paper-lined baking sheet. Brush with egg white. Sprinkle 1/2 cup cheese in center of crust. Spoon onion mixture over cheese, leaving a 2 1/2-inch border. Sprinkle remaining 1/4 cup cheese over onion. Fold piecrust border up and over onion, pleating as you go and leaving a 4-inch-wide opening in center. Brush crust with egg white.
Bake at 425° on bottom oven rack 17 to 19 minutes or until crust is golden. Let stand 5 minutes before serving.
Rustic Vidalia Onion Tart With Thyme: Substitute chopped fresh thyme for rosemary. Proceed with recipe as directed.